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Found in: Page 289

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Consider a ${\mathbf{4}}{\mathbf{×}}{\mathbf{4}}$ matrix A with rows ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{1}}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{2}}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{3}}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{4}}}$. If , find the determinants in Exercises 11 through 16.12. localid="1659509477853" ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\mathbf{\left[}\begin{array}{c}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{1}}\\ {\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{2}}\\ {\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{3}}\\ {\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{4}}\end{array}\mathbf{\right]}$

Therefore, the determinant of given determinant matrix is given by,

$det\left[\begin{array}{c}{\stackrel{\to }{v}}_{1}\\ {\stackrel{\to }{v}}_{2}\\ {\stackrel{\to }{v}}_{3}\\ {\stackrel{\to }{v}}_{4}\end{array}\right]=-8$

See the step by step solution

## Step 1: Definition

A determinant is a unique number associated with a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

## Step 2: Given

Given determinant,

$\mathrm{det}\left(\mathrm{A}\right)=8\phantom{\rule{0ex}{0ex}}\mathrm{det}\left[\begin{array}{c}{\stackrel{\to }{\mathrm{v}}}_{1}\\ {\stackrel{\to }{\mathrm{v}}}_{2}\\ {\stackrel{\to }{\mathrm{v}}}_{3}\\ {\stackrel{\to }{\mathrm{v}}}_{4}\end{array}\right]$

## Step 3: To find determinant

To determine,

$det\left[\begin{array}{c}{\stackrel{\to }{v}}_{1}\\ {\stackrel{\to }{v}}_{2}\\ {\stackrel{\to }{v}}_{3}\\ {\stackrel{\to }{v}}_{4}\end{array}\right]=-det\left[\begin{array}{c}{\stackrel{\to }{v}}_{1}\\ {\stackrel{\to }{v}}_{2}\\ {\stackrel{\to }{v}}_{3}\\ {\stackrel{\to }{v}}_{4}\end{array}\right]\phantom{\rule{0ex}{0ex}}det\left[\begin{array}{c}{\stackrel{\to }{v}}_{1}\\ {\stackrel{\to }{v}}_{2}\\ {\stackrel{\to }{v}}_{3}\\ {\stackrel{\to }{v}}_{4}\end{array}\right]=-det\left(A\right).\phantom{\rule{0ex}{0ex}}det\left[\begin{array}{c}{\stackrel{\to }{v}}_{1}\\ {\stackrel{\to }{v}}_{2}\\ {\stackrel{\to }{v}}_{3}\\ {\stackrel{\to }{v}}_{4}\end{array}\right]=-8$